A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

Let K be any field, K(x1, . . . , xn) be the rational function field of n variables over K, and Sn and An be the symmetric group and the alternating group of degree n respectively. For any a ∈ K \ {0}, define an action of Sn on K(x1, . . . , xn) by σ · xi = xσ(i) for σ ∈ An and σ · xi = a/xσ(i) for σ ∈ Sn \ An. Theorem. For any field K and n = 3, 4, 5, the fixed field K(x1, . . . , xn) Sn is ra...

for a composition $lambda$ of $n$ our aim is to obtain reduced forms ‎for all the elements in the kazhdan-lusztig (right) cell containing ‎$w_{j(lambda)}$‎, ‎the longest element of the standard parabolic‎ ‎subgroup of $s_n$ corresponding to $lambda$‎. ‎we investigate how far this is possible to achieve by looking at‎ ‎elements of the form $w_{j(lambda)}d$‎, ‎where $d$ is a prefix of ‎an element...